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On Designing Mixed Nonparametric Control Chart for Monitoring the Manufacturing Processes

  • Research Article-Mechanical Engineering
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Abstract

Efficient control charts are used to maintain and improve the manufacturing, industrial and service processes. Control chart is a basic tool of quality practitioners for producing products that meet market requirements in terms of quality, reliability and functionality. Lack of normality is a common occurrence in these processes. In such situations, distribution-free control charts are increasingly used for process monitoring. The nonparametric exponentially weighted moving average chart is a frequently used memory-type control chart in the process monitoring. The drawback of this chart is that it performs efficiently only on smaller values of the smoothing parameter. To compensate this drawback, a modified nonparametric exponentially weighted moving average chart under progressive setup based on sign and arcsine test statistics is proposed in this study. The prominent quality of the proposed scheme is that it performs efficiently in detecting small and persistent shifts in the process location under each choice of the smoothing parameter. The performance of the proposed chart has been investigated through simulations that use run length profiles (average run length, median run length and standard deviation of run length). When the performance of the proposed chart is compared with alternatives, its ability to detect small and persistent shifts is much better. Two real-life applications associated with hard-bake and piston rings manufacturing processes are included that show the demonstration of the proposed charts.

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Acknowledgements

The aforementioned work was partially supported by National Natural Science Foundation of China (No. 11731015 and 11571148). The authors are grateful to the editor and referees for their constructive comments that led to substantial improvements in the article.

Author information

Authors and Affiliations

  1. School of Economics and Statistics, Guangzhou University, Guangzhou, China

    Saber Ali, Xingfa Zhang & Yuan Li

  2. Department of Statistics, Government Ambala Muslim Graduate College Sargodha, Sargodha, Pakistan

    Zameer Abbas

  3. Department of Statistics, University of Sargodha, Sargodha, Pakistan

    Hafiz Zafar Nazir

  4. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Muhammad Riaz

Authors
  1. Saber Ali
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  2. Zameer Abbas
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  3. Hafiz Zafar Nazir
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  4. Muhammad Riaz
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  5. Xingfa Zhang
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  6. Yuan Li
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Correspondence to Xingfa Zhang.

Appendices

Appendix

NPMEP chart is the mixture of NPEWMA and nonparametric PM chart. To derive the mean and variance of NPMEP chart, first we must know about the mean and variance of NPEWMA and nonparametric PM chart.

Appendix-A

Mean and variance of NPEWMA chart under sign test statistic. The NPEWMA statistic defined in Eq. (2) can be written as:

$$E_{{S_{t}^{ + } }} = \omega \mathop \sum \limits_{j = 1}^{t - 1} \left( {1 - \omega } \right)^{j} S_{t - j}^{ + } + \left( {1 - \omega } \right)^{t} E_{{S_{t = 0}^{ + } }} ,$$
$$\begin{aligned} E(E_{{S_{t}^{ + } }} ) = & E\left[ { \omega \mathop \sum \limits_{j = 1}^{t - 1} \left( {1 - \omega } \right)^{j} S_{t - j}^{ + } + \left( {1 - \omega } \right)^{t} E_{{S_{t = 0}^{ + } }} } \right], \\ = & \left[ { \omega \left( {1 - \left( {1 - \omega } \right)^{t} } \right)/\omega \left( {np_{0} } \right) + \left( {1 - \omega } \right)^{t} np_{0} } \right], \\ = & np_{0} \left[ { \omega \left( {1 - \left( {1 - \omega } \right)^{t} } \right)/\omega + \left( {1 - \omega } \right)^{t} } \right], \\ = & np_{0} \\ \end{aligned}$$
(A1)
$$\begin{aligned} {\text{var}} (E_{{S_{t}^{ + } }} ) = & {\text{var}} \left[ { \omega \mathop \sum \limits_{j = 1}^{t - 1} \left( {1 - \omega } \right)^{j} S_{t - j}^{ + } + \left( {1 - \omega } \right)^{t} E_{{S_{t = 0}^{ + } }} } \right], \\ = & np_{0} \left( {1 - P_{0} } \right)\omega^{2} \left[ {\left( {1 - \left( {1 - \omega } \right)^{2t} } \right)/\omega \left( {2 - \omega } \right)^{t} } \right], \\ = & \frac{\omega }{2 - \omega }\left( {1 - \left( {1 - \omega } \right)^{2t} } \right)\left( {np_{0} \left( {1 - p_{0} } \right)} \right) \\ \end{aligned}$$
(A2)

where \(p_{0} = 1/2\).

Appendix-B

Mean and variance of nonparametric PM chart. The nonparametric PM statistic defined in Eq. (5) is given as:

\({\text{NPPM}}_{{S_{t} }} = \frac{{S_{1}^{ + } + S_{2}^{ + } + \cdots + S_{t}^{ + } }}{t}\),

$$\begin{aligned} E({\text{NPPM}}_{{S_{t} }} ) = & E\left[ {\frac{{S_{1}^{ + } + S_{2}^{ + } + \cdots + S_{t}^{ + } }}{t}} \right] \\ = & \frac{{E\left( {S_{1}^{ + } } \right) + E\left( {S_{2}^{ + } } \right) + \cdots + E\left( {S_{t}^{ + } } \right)}}{t}, \\ = & \frac{{np_{0} + np_{0} + \cdots + np_{0} }}{t}, \\ = & np_{0} \\ \end{aligned}$$
(B1)
$$\begin{aligned} {\text{var}} ({\text{NPPM}}_{{S_{t} }} ) = & {\text{var}} \left[ {\frac{{S_{1}^{ + } + S_{2}^{ + } + \cdots + S_{t}^{ + } }}{t}} \right] \\ = & \frac{{{\text{var}} \left( {S_{1}^{ + } } \right) + {\text{var}} \left( {S_{2}^{ + } } \right) + \cdots + {\text{var}} \left( {S_{t}^{ + } } \right)}}{{t^{2} }}, \\ = & \frac{{np_{0} \left( {1 - p_{0} } \right) + np_{0} \left( {1 - p_{0} } \right) + \cdots + np_{0} \left( {1 - p_{0} } \right)}}{{t^{2} }}, \\ = & \frac{{np_{0} \left( {1 - p_{0} } \right)}}{t} \\ \end{aligned}$$
(B2)

Appendix-C

Mean and variance of the proposed NPMEP sign chart is derived as follows:

$$NPMEP_{{S_{t} }} = \frac{1}{t}\mathop \sum \limits_{j = 1}^{t} E_{{S_{j}^{ + } }} .$$
$$E({\text{NPMEP}}_{{S_{t} }} ) = E\left[ {\frac{1}{t}\mathop \sum \limits_{j = 1}^{t} E_{{S_{j}^{ + } }} } \right],$$
$$E({\text{NPMEP}}_{{S_{t} }} ) = \frac{1}{t}\left[ {\mathop \sum \limits_{j = 1}^{t} E(E_{{S_{j}^{ + } }} )} \right],$$
$$E({\text{NPMEP}}_{{S_{t} }} ) = t\frac{{np_{0} }}{t},$$
$$E({\text{NPMEP}}_{{S_{t} }} ) = np_{0} .$$
(C1)

For variance of the NPMEP statistic.

\(E_{{S_{j}^{ + } }} = \omega \mathop \sum \nolimits_{i = 0}^{j - 1} \left( {1 - \omega } \right)^{i} S_{t - j}^{ + } + \left( {1 - \omega } \right)^{j} E_{0}\), where \(E_{0}\) is the initial value which is equal to the mean of the NPEWMA statistic.

The NPMEP statistic

$${\text{NPMEP}}_{{S_{t} }} = \frac{{\mathop \sum \nolimits_{j = 1}^{t} E_{{S_{j}^{ + } }} }}{t}$$
$${\text{NPMEP}}_{{S_{t} }} = \frac{{\mathop \sum \nolimits_{j = 1}^{t} \omega \mathop \sum \nolimits_{i = 0}^{j - 1} \left( {1 - \omega } \right)^{i} S_{j - i}^{ + } + \left( {1 - \omega } \right)^{j} E_{0} }}{t}$$
$${\text{NPMEP}}_{{S_{t} }} = \frac{\omega }{t}\mathop \sum \limits_{j = 1}^{t} \mathop \sum \limits_{i = 0}^{j - 1} \left( {1 - \omega } \right)^{i} S_{j - i}^{ + } + \frac{1}{t}\mathop \sum \limits_{j = 1}^{t} \left( {1 - \omega } \right)^{j} E_{0}$$

Consider

$$\mathop \sum \limits_{j = 1}^{t} \mathop \sum \limits_{i = 0}^{j - 1} \left( {1 - \omega } \right)^{i} S_{j - i}^{ + } ,$$
$$= \left[ {\mathop \sum \limits_{i = 0}^{1 - 1} \left( {1 - \omega } \right)^{i} S_{1 - i}^{ + } } \right] + \left[ {\mathop \sum \limits_{i = 0}^{2 - 1} \left( {1 - \omega } \right)^{i} S_{2 - i}^{ + } } \right] + \cdots + \left[ {\mathop \sum \limits_{i = 0}^{t - 1} \left( {1 - \omega } \right)^{i} S_{t - i}^{ + } } \right],$$

Open these summations and then we get

$$\mathop \sum \limits_{j = 1}^{t} \left( {\mathop \sum \limits_{i = 0}^{t - j} \left( {1 - \omega } \right)^{i} } \right)S_{j}^{ + }$$
$${\text{NPMEP}}_{{S_{t} }} = \frac{\omega }{t}\;\mathop \sum \limits_{j = 1}^{t} \left( {\mathop \sum \limits_{i = 0}^{t - j} \left( {1 - \omega } \right)^{i} } \right)S_{j}^{ + } + \frac{1}{t}\mathop \sum \limits_{j = 1}^{t} \left( {1 - \omega } \right)^{j} E_{0}$$

For variance

$${\text{var}} \left( {{\text{NPMEP}}_{{S_{t} }} } \right) = {\text{var}} \left( {\frac{\omega }{t} \mathop \sum \limits_{j = 1}^{t} \left( {\mathop \sum \limits_{i = 0}^{t - j} \left( {1 - \omega } \right)^{i} } \right)S_{j}^{ + } + \frac{1}{t}\mathop \sum \limits_{j = 1}^{t} \left( {1 - \omega } \right)^{j} E_{0} } \right),$$

\(= \frac{{\omega^{2} }}{{t^{2} }}\) \(\left( {\mathop \sum \limits_{j = 1}^{t} \left( {\mathop \sum \limits_{i = 0}^{t - j} \left( {1 - \omega } \right)^{i} } \right)} \right)^{2} \frac{n}{4} + 0;\) as \({\text{var}} \left( {\frac{1}{t}\mathop \sum \limits_{j = 1}^{t} \left( {1 - \omega } \right)^{j} E_{0} } \right) = 0,\)

$$= \frac{{\omega^{2} }}{{t^{2} }}\;\mathop \sum \limits_{j = 1}^{t} \frac{{\left( {1 - \left( {1 - \omega } \right)^{t - j + 1} } \right)}}{{\omega^{2} }}^{2} \left( \frac{n}{4} \right),$$
$$= \frac{n}{{4t^{2} }}\mathop \sum \limits_{j = 1}^{t} \left( {1 - \left( {1 - \omega } \right)^{t - j + 1} } \right)^{2}$$
(C2)

Using geometric progression to simplify the expression given in (C2)

$$= \frac{n}{{4t^{2} }}\;\mathop \sum \limits_{j = 1}^{t} [1 + \left( {1 - \omega } \right)^{{2\left( {t - j + 1} \right)}} - 2\left( {1 - \omega } \right)^{t - j + 1} ]$$
$$= \frac{n}{{4t^{2} }}\left[ {\mathop \sum \limits_{j = 1}^{t} 1 + \mathop \sum \limits_{j = 1}^{t} \left( {1 - \omega } \right)^{{2\left( {t - j + 1} \right)}} - 2\mathop \sum \limits_{j = 1}^{t} \left( {1 - \omega } \right)^{t - j + 1} } \right],$$

Appling sum of the geometric progressions

$$= \frac{n}{{4t^{2} }}\left[ {t + \left( {1 - \omega } \right)^{2} \left( {\frac{{1 - \left( {1 - \omega } \right)^{2t} }}{{1 - \left( {1 - \omega } \right)^{2} }}} \right) - 2\left( {1 - \omega } \right)\left( {\frac{{1 - \left( {1 - \omega } \right)^{t} }}{\omega }} \right)} \right],$$
$$= \frac{nt}{{4t^{2} }}\left[ {1 + \frac{{\left( {1 - \omega } \right)^{2} }}{t}\left( {\frac{{1 - \left( {1 - \omega } \right)^{2t} }}{{1 - \left( {1 - \omega } \right)^{2} }}} \right) - 2\frac{{\left( {1 - \omega } \right)}}{t}\left( {\frac{{1 - \left( {1 - \omega } \right)^{t} }}{\omega }} \right)} \right],$$
$${\text{var}} \left( {{\text{NPMEP}}_{{S_{t} }} } \right) = \frac{n}{4t}\left[ {1 + \frac{{\left( {1 - \omega } \right)^{2} }}{t}\left( {\frac{{1 - \left( {1 - \omega } \right)^{2t} }}{{1 - \left( {1 - \omega } \right)^{2} }}} \right) - 2\frac{{\left( {1 - \omega } \right)}}{t}\left( {\frac{{1 - \left( {1 - \omega } \right)^{t} }}{\omega }} \right)} \right].$$
(C3)

Appendix-D

Mean and variance of the proposed NPMEP arcsine chart.

\(E({\text{NPMEP}}_{{M_{t} }} ) = \mathop \sum \limits_{j = 1}^{t} \frac{{E\left( {E_{{M_{j} }} } \right)}}{t},\),

$$E({\text{NPMEP}}_{{M_{t} }} ) = t\frac{{\sin^{ - 1} \sqrt {P_{0} } }}{t},$$
$$E({\text{NPMEP}}_{{M_{t} }} ) = \sin^{ - 1} \sqrt {P_{0} } .$$
(D1)

For variance of \({\text{NPMEP}}_{{M_{t} }}\)

$${\text{NPMEP}}_{{M_{t} }} = \frac{{\mathop \sum \nolimits_{j = 1}^{t} \omega \mathop \sum \nolimits_{i = 0}^{j - 1} \left( {1 - \omega } \right)^{i} M_{j - i} + \left( {1 - \omega } \right)^{j} E_{0} }}{t}$$
$${\text{NPMEP}}_{{M_{t} }} = \frac{\omega }{t}\mathop \sum \limits_{j = 1}^{t} \mathop \sum \limits_{i = 0}^{j - 1} \left( {1 - \omega } \right)^{i} M_{j - i} + \frac{1}{t}\mathop \sum \limits_{j = 1}^{t} \left( {1 - \omega } \right)^{j} E_{0}$$

Consider

$$\mathop \sum \limits_{j = 1}^{t} \mathop \sum \limits_{i = 0}^{j - 1} \left( {1 - \omega } \right)^{i} M_{j - i} ,$$
$$= \left[ {\mathop \sum \limits_{i = 0}^{1 - 1} \left( {1 - \omega } \right)^{i} M_{1 - i} } \right] + \left[ {\mathop \sum \limits_{i = 0}^{2 - 1} \left( {1 - \omega } \right)^{i} M_{2 - i} } \right] + \ldots + \left[ {\mathop \sum \limits_{i = 0}^{t - 1} \left( {1 - \omega } \right)^{i} M_{t - i} } \right],$$

Open these summations and then we get

$$\mathop \sum \limits_{j = 1}^{t} \left( {\mathop \sum \limits_{i = 0}^{t - j} \left( {1 - \omega } \right)^{i} } \right)M_{j}$$
$${\text{NPMEP}}_{{S_{t} }} = \frac{\omega }{t}\;\mathop \sum \limits_{j = 1}^{t} \left( {\mathop \sum \limits_{i = 0}^{t - j} \left( {1 - \omega } \right)^{i} } \right)M_{j} + \frac{1}{t}\mathop \sum \limits_{j = 1}^{t} \left( {1 - \omega } \right)^{j} E_{0}$$

For variance

$${\text{var}} \left( {{\text{NPMEP}}_{{S_{t} }} } \right) = {\text{var}} \left( {\frac{\omega }{t} \mathop \sum \limits_{j = 1}^{t} \left( {\mathop \sum \limits_{i = 0}^{t - j} \left( {1 - \omega } \right)^{i} } \right)M_{j} + \frac{1}{t}\mathop \sum \limits_{j = 1}^{t} \left( {1 - \omega } \right)^{j} E_{0} } \right),$$

\(= \frac{{\omega^{2} }}{{t^{2} }}\) \(\left( {\mathop \sum \limits_{j = 1}^{t} \left( {\mathop \sum \limits_{i = 0}^{t - j} \left( {1 - \omega } \right)^{i} } \right)} \right)^{2} \frac{1}{4n} + 0\) as \({\text{var}} \left( {\frac{1}{t}\mathop \sum \limits_{j = 1}^{t} \left( {1 - \omega } \right)^{j} E_{0} } \right) = 0\)

$$= \frac{{\omega^{2} }}{{t^{2} }}\;\mathop \sum \limits_{j = 1}^{t} \frac{{\left( {1 - \left( {1 - \omega } \right)^{t - j + 1} } \right)}}{{\omega^{2} }}^{2} \left( \frac{1}{4n} \right)$$
$$= \frac{1}{{4nt^{2} }}\mathop \sum \limits_{j = 1}^{t} \left( {1 - \left( {1 - \omega } \right)^{t - j + 1} } \right)^{2}$$
$$= \frac{1}{{4nt^{2} }}\;\mathop \sum \limits_{j = 1}^{t} [1 + \left( {1 - \omega } \right)^{{2\left( {t - j + 1} \right)}} - 2\left( {1 - \omega } \right)^{t - j + 1} ]$$
$$= \frac{1}{{4nt^{2} }}\left[ {\mathop \sum \limits_{j = 1}^{t} 1 + \mathop \sum \limits_{j = 1}^{t} \left( {1 - \omega } \right)^{{2\left( {t - j + 1} \right)}} - 2\mathop \sum \limits_{j = 1}^{t} \left( {1 - \omega } \right)^{t - j + 1} } \right],$$

Appling sum of the geometric progressions

$$= \frac{1}{{4nt^{2} }}\left[ {t + \left( {1 - \omega } \right)^{2} \left( {\frac{{1 - \left( {1 - \omega } \right)^{2t} }}{{1 - \left( {1 - \omega } \right)^{2} }}} \right) - 2\left( {1 - \omega } \right)\left( {\frac{{1 - \left( {1 - \omega } \right)^{t} }}{\omega }} \right)} \right],$$
$$= \frac{t}{{4nt^{2} }}\left[ {1 + \frac{{\left( {1 - \omega } \right)^{2} }}{t}\left( {\frac{{1 - \left( {1 - \omega } \right)^{2t} }}{{1 - \left( {1 - \omega } \right)^{2} }}} \right) - 2\frac{{\left( {1 - \omega } \right)}}{t}\left( {\frac{{1 - \left( {1 - \omega } \right)^{t} }}{\omega }} \right)} \right],$$
$${\text{var}} \left( {{\text{NPMEP}}_{{S_{t} }} } \right) = \frac{1}{4nt}\left[ {1 + \frac{{\left( {1 - \omega } \right)^{2} }}{t}\left( {\frac{{1 - \left( {1 - \omega } \right)^{2t} }}{{1 - \left( {1 - \omega } \right)^{2} }}} \right) - 2\frac{{\left( {1 - \omega } \right)}}{t}\left( {\frac{{1 - \left( {1 - \omega } \right)^{t} }}{\omega }} \right)} \right].$$
(D2)

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Ali, S., Abbas, Z., Nazir, H.Z. et al. On Designing Mixed Nonparametric Control Chart for Monitoring the Manufacturing Processes. Arab J Sci Eng 46, 12117–12136 (2021). https://doi.org/10.1007/s13369-021-05801-6

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